Pareto Distribution

Welcome to The Coding College, your go-to resource for simplifying data science and programming concepts! In this guide, we’ll explore the Pareto Distribution, its significance, properties, applications, and how to implement it in Python using NumPy.

What is the Pareto Distribution?

The Pareto Distribution, also known as the Power Law Distribution, is a continuous probability distribution that describes phenomena where a small proportion of causes contributes to the majority of effects. This is often referred to as the 80/20 rule: 80% of outcomes come from 20% of causes.

Probability Density Function (PDF):

The PDF of the Pareto Distribution is given by:

Where:

• xm: The minimum value (scale parameter, x>0).
• α\alpha: The shape parameter (>0).

Key Characteristics

Real-Life Applications

1. Wealth Distribution: A small percentage of individuals hold most of the wealth.
2. Natural Phenomena: Earthquake magnitudes, city populations, and forest fire sizes.
3. Business: Product sales where a few products generate most of the revenue.
4. Internet Traffic: A small percentage of users generate most of the traffic.

Pareto Distribution in NumPy

Python’s NumPy library provides a function to generate random samples from the Pareto distribution:

Syntax:

numpy.random.pareto(a, size=None)

• a: Shape parameter (α\alpha).
• size: Output shape (default is None, which returns a single value).

Example 1: Generating Random Numbers

Scenario: Simulate wealth distribution

import numpy as np

# Generate Pareto random numbers
alpha = 3  # Shape parameter
data = np.random.pareto(a=alpha, size=10) + 1  # Add 1 to include the minimum value
print("Random samples from Pareto distribution:", data)

Output (Example):

[1.54 1.21 1.13 1.75 1.32 1.89 1.05 1.67 1.23 1.45]

Example 2: Visualizing the Pareto Distribution

import numpy as np
import matplotlib.pyplot as plt

# Generate data
alpha = 2.0
data = np.random.pareto(a=alpha, size=1000) + 1  # Add 1 for the scale

# Plot histogram
plt.hist(data, bins=50, color='skyblue', edgecolor='black', density=True)
plt.title('Pareto Distribution (α=2)')
plt.xlabel('Value')
plt.ylabel('Density')
plt.grid(True)
plt.show()

Example 3: Comparing Pareto Distributions

Scenario: Analyze the effect of the shape parameter

import numpy as np
import matplotlib.pyplot as plt

# Generate data with different shape parameters
data1 = np.random.pareto(a=1.5, size=1000) + 1
data2 = np.random.pareto(a=3.0, size=1000) + 1
data3 = np.random.pareto(a=5.0, size=1000) + 1

# Plot histograms
plt.hist(data1, bins=50, alpha=0.5, label='α=1.5', density=True, color='blue')
plt.hist(data2, bins=50, alpha=0.5, label='α=3.0', density=True, color='orange')
plt.hist(data3, bins=50, alpha=0.5, label='α=5.0', density=True, color='green')

plt.title('Pareto Distributions with Different Shape Parameters')
plt.xlabel('Value')
plt.ylabel('Density')
plt.legend()
plt.grid(True)
plt.show()

Example 4: Transforming the Pareto Distribution

You can customize the Pareto Distribution by multiplying by a scale factor (xmx_m):

import numpy as np

# Shape and scale parameters
alpha = 2.5
x_m = 2  # Minimum value (scale parameter)

# Generate Pareto random numbers
data = (np.random.pareto(a=alpha, size=1000) + 1) * x_m

# Print sample statistics
print("Mean:", np.mean(data))
print("Variance:", np.var(data))

Properties of the Pareto Distribution

Pareto vs Other Distributions

Summary

The Pareto Distribution is a versatile tool for modeling power-law phenomena in various fields